ABSTRACT The deep nuclei of the cerebellar cortex have not yet received adequate exploratory attention. An exception is represented by the pioneering work of Chan-Palay, published in 1977, on the dentate nucleus morphology. She has classified each individual cell in the dentatus of the monkey into one of six types. Although fractal analysis is presently the most prominent quantitative method for morphometric neuronal studies, no article referring to applications of this method to the analysis of cell types of the dentate nucleus has so far been published. In the present study we apply fractal analysis to this unsolved problem and calculate the fractal dimension for each dendritic arbour of a neuron. We will hereby prove that by application of fractal analysis to the dendritic arbours of these cells whilst ignoring other neuronal attributes allows for clear discrimination of only three cell types.

[Show abstract][Hide abstract]ABSTRACT: The dentate nucleus represents the most lateral of the four cerebellar nuclei that serve as major relay centres for fibres coming from the cerebellar cortex. Although many relevant findings regarding to the structure, neuronal morphology and cytoarchitectural development of the dentate nucleus have been presented so far, very little quantitative information has been collected on the types of large neurons in the human dentate nucleus. In the present study we qualitatively analyze our sample of large neurons according to their morphology and topology, and classify these cells into four types. Then, we quantify the morphology of such cell types taking into account seven morphometric parameters which describe the main properties of the cell soma, dendritic field and dendrite arborization. By performing appropriate statistics we prove out our classification of the large dentate neurons in the adult human. To the best of our knowledge, this study represents the first attempt of quantitative analysis of morphology and classification of the large neurons in the adult human dentate nucleus.

[Show abstract][Hide abstract]ABSTRACT: The morphology of neurons from the human dentate nucleus was analyzed estimating the size and shape of the dendritic field, shape of the neuron, space-filling property and the degree of dendrite aberrations. Among them, the last three morphological properties were investigated using the most popular technique of fractal analysis: the box-count method. The box dimensions of binary images and dendritic field area were statistically investigated in order to test whether the binary box dimension can quantify the size of the neuron. The same analysis was carried out using the box dimension of outline images and image circularity. The parameters, presented in this study have proved to be a useful means for quantifying the morphology of dentate neurons as they provide a robust means of differentiating between neuron subtypes in the dentate nucleus. The findings of the present study are in accordance with previous qualitative data.

Proceedings of the 2013 19th International Conference on Control Systems and Computer Science; 05/2013

[Show abstract][Hide abstract]ABSTRACT: This aim of this study was to assess the discriminatory value of fractal and grey level co-occurrence matrix (GLCM) analysis methods in standard microscopy analysis of two histologically similar brain white mass regions that have different nerve fiber orientation. A total of 160 digital micrographs of thionine-stained rat brain white mass were acquired using a Pro-MicroScan DEM-200 instrument. Eighty micrographs from the anterior corpus callosum and eighty from the anterior cingulum areas of the brain were analyzed. The micrographs were evaluated using the National Institutes of Health ImageJ software and its plugins. For each micrograph, seven parameters were calculated: angular second moment, inverse difference moment, GLCM contrast, GLCM correlation, GLCM variance, fractal dimension, and lacunarity. Using the Receiver operating characteristic analysis, the highest discriminatory value was determined for inverse difference moment (IDM) (area under the receiver operating characteristic (ROC) curve equaled 0.925, and for the criterion IDM≤0.610 the sensitivity and specificity were 82.5 and 87.5%, respectively). Most of the other parameters also showed good sensitivity and specificity. The results indicate that GLCM and fractal analysis methods, when applied together in brain histology analysis, are highly capable of discriminating white mass structures that have different axonal orientation.

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Author's personal copycorresponding line of regression, and measuring the degree ofexperimental data points’ fluctuation around this line.The images of Golgi impregnated neurons of the cat spinalcord were taken from the experimental data published in threeoriginal articles [4,5,12]. A detailed description of the adoptedhistological procedure can be found in these references. Theimages of all neurons were classified according to Rexed’s lam-D. Ristanovi´ c, N.T. Miloˇ sevi´ c / Neuroscience Letters 414 (2007) 286–290287be reformulated (if possible) to accommodate for the results ofsuch experiment.The significance of Rexed’s lamination is based on a purelycytoarchitectonic approach (pertaining only to the cellulararrangement within laminae), neglecting one of the principalmorphologic attributes of the neuron—dendritic tree geometry[18]. It should be emphasized that this geometry represents animportant factor in all neuronal classifications, as cytoarchitec-tonicstudieshavenotleadtowardsamoredetailedclassificationof spinal neurons. Since Rexed’s descriptions are based almostentirely upon the size, location, and grouping of cell bodies asrevealed by Nissl stain, and since such a nuclear grouping bearslittlediscernablerelationtodendriticfields,ourstudyhasmainlybeen oriented towards reviewing the dendritic organization ofthe neurons from the dorsal horn. As such, the purpose of thepresent study was to evaluate Rexed’s hypothesis by testing thestructure of dendritic arborization patterns of cells from the catspinal cord using the linear Sholl method, complemented withparametric and nonparametric statistics to cover the differencesin complexity of dendritic trees between neurons taken fromdifferent laminae.The Sholl’s analysis [19] is a commonly used quantitativemethod specifying dendritic geometry, ramification richness,and dendritic branching pattern. This analysis consists of (i) theconstruction of concentric and regularly spaced circles centeredonthecellbody,(ii)thecountofintersectionsofdendritescross-ing each of the circles of increasing radii, and (iii) the choice ofappropriate mathematical techniques for data presentation andprocessing (i.e., histograms, Schoenen’s vectograms, or plots incoordinate systems). The linear Sholl method is a basic mod-ification of the Sholl analysis geared towards analyzing therelationships between the numbers of dendritic intersectionswith the circles versus the circle radius. However, the Shollanalysis is traditionally carried out by using either semi-log,or log-log data method. We have argued that the linear and log-arithmic methods are complementary (i.e. they should be usedtogether) since each offers different information on neuronalarborization patterns [15]. Data obtained using the linear anal-ysis is rarely subjected to a complete statistical analysis. Someauthors often use standard population parameters, and samplevariables of parametric statistics, such as the mean, the standarddeviation,anddifferenttestsofsignificance,withoutevertestingtheir data for normality. The study we are presenting has beenspecifically designed to test data for normality prior to choosingthe appropriate data processing method.In the linear Sholl analysis, it is important to represent thefrequency plots properly. Current literature does not show pre-cisely how these plots were obtained. Thus, we find it necessaryto include the problem related to finding the equation of theline representing the “best fit” to a scatter diagram, plotting theinar scheme and divided into groups based on their laminarposition, in such manner that 8 neurons are chosen from laminaI,17neuronsarechosenfromthesubstantiagelatinosa(laminaeII and III), 16 neurons are chosen from lamina IV, 15 neuronsare chose from lamina V, and finally 8 neurons are chosen fromlamina VI. The drawings of these neurons were converted intodigitizedimagesusingascannerwitharesolutionof600dpi.Alltransformations were carried out on a computer using the publicdomain Image J software (www.rcb.info.nih.gov/ij) developedat the US National Institute of Health. All scanned images wereimported into the software. Axons, spines (if present), and thecellbodyhavebeendigitallyremovedfromthescanneddrawing.Each dendrite was filled with pixels. Drawings were then ana-lyzed as “skeleton” tracings. To that purpose the same softwareperformed the transformation of the image into a stick figure.In order to study the variation of the number of neuronalbranches with the distance from the cell body, we found it con-venient to use, after Sholl, a series of concentric circles with acommoncentreinthecellbody,ascoordinatesofreference,andtraced the network of such circles over a single neuronal image,with radii increasing at regular steps of 15?m in all cases. Forthese consecutive r-values, we chose 10, 25, 40, 55, ..., ?m.The whole analysis, including the drawing of the network ofcircles,andcountingtheintersectionsNofthesecircleswiththedendrites, was done using Image J software.In order to use a Chi-square distribution to test a populationfornormality,afairlylargesampleofthepopulationisnecessary(e.g., the ordinates of the points drawn in r,N-plane). If samplesare smaller, the third (m3) and fourth (m4) empirical momentshould be estimated. One of the measures describing the asym-metry of a distribution about a maximum is called the skewness(a3), and a measure of the degree of the kurtosis (a4) of the dis-tribution is called the excess (e). They are given, respectively,bya3=m3s3,e = a4− 3 =m4s4− 3,(1)where s is the empirical standard deviation. The value of theskewness for a perfectly symmetrical distribution of data is 0(null), and the excess for the normal curve also equals 0. Theseparameters should be very small in the case of an experimentalnormal distribution. How small those parameters should be toenable the normality of the distribution can be estimated bydividing the corresponding mean square errors?σ3=6(n − 1)(n + 1)(n + 3),σ4=?24n(n − 2)(n − 3)(n − 1)2(n + 3)(n + 5)(2)with the values given by Eq. (1), where n is the number ofvariables in the sample. If????[7]. This procedure requires each of the expected frequencies beat least 5.σ3a3????≤ 2.0,???σ4e??? ≤ 2.0 (3)there is ground to claim that the data are normally distributed

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Author's personal copya vertical straight line defined by the equation N=ri. This posestheissuewhetherthesedatapointsarenormallydistributedalongthis straight line. If that is the case, we can use the standard val-ues from parametric statistics, such as, the mean, the standarderror, and so on. If the number of these variables (m) is not lessthan 5, we can apply for each r (10, 25, 40, ... ?m) separatelythe above shown test for normality and check if these samples288D. Ristanovi´ c, N.T. Miloˇ sevi´ c / Neuroscience Letters 414 (2007) 286–290When there is some indication of the distribution characterin a population, it is often possible to fit such theoretical distri-bution (a fitted curve) to a frequency distribution obtained froma sample of the population. If a scatter diagram in the r,N-planeindicates that a straight line will not fit a set of points becauseof the nonlinearity of the relationship between N and r, it isstill possible that there is some other type of relationship. Asthere is a tendency to provide as simple as possible explanationsof relationships, polynomials are usually selected as frequencyfunctions because of their simplicity and flexibility [1].Let the degree of the polynomial be k and let the polynomialrelationship of degree k in r be written in the formN = a0+ a1r + a2r2+ ... + akrk.Here, the problem is to determine from a given sample of npairs of r,N-values (ri,Ni) the (k+1) values of the constants a0,a1, a2, ..., ak. By means of the differential calculus and leastsquares method, it can be shown [1] that these constants mustbethesolutionsofthefollowingsystemofk+1algebraiclinearequations (also known as normal equations):?r + a2a0···a0(4)a0n + a1?r + a1?rk+a1Since in Eq. (5) confusion is not possible, the summationindex i is omitted. The polynomial of “best fit” will be found bysolving the system of Eq. (5) for aj(j=0, 1, 2, ..., k) and sub-stituting the obtained values into Eq. (4). This system of linearequations can easily be solved by using the package Mathemat-ica (Wolfram).We consider the cat spinal cord as a system S constituted ofspinal neurons as the elements. For each element of this systemwe define N(r) as the number of intersections of neuronal den-drites with a circle of a given radius r. This quantity defines therate of dendritic branching at increasing distances from the cellbody. After Rexed, we suppose that system S is divided into ksubsystems Sj(j=1, 2, ..., k) (in our case, into four laminaeand one laminar group—the substantia gelatinosa). For everyneuron of a given lamina Sjthe intersections of dendrites witheach particular circle from the series of concentric and regularlyspaced circles were counted and all these data were presented inthe r,N-planes. Since data were obtained for the same series ofthe independent variable r (that is, for 10, 25, 40, ..., ?m), thepoints for the particular value of ri, whose ordinates are N1, N2,..., Nm(m≤n since for a given radius risome of the neuronaldendritic fields could be smaller than others), are positioned on?r2+ ··· + ak?rk=?N?r2+ a2?rk+1+ a2?r3+ ··· + ak?rk+2+ ··· + ak?rk+1=?rN?r2k=?rkN(5)Fig. 1. Illustration of an application of the Sholl linear method to neurons fromthe substantia gelatinosa from a dorsal horn of the cat spinal cord. Each point isthemeanofthenumberofintersections(N)versusthecircleradius(r)calculatedfor all measured neurons. For clarity some of the data points are omitted. Thebars represent the standard errors. The graph shown by broken line is obtainedby fitting the polynomial of degree 4 to data points (the correlation coefficientR=0.98) and the graph represented by the full line (Int) is calculated usingLagrange’s interpolation polynomial of degree 9 (R=1).N1, N2, ..., Nmprove that the population of the data satisfies anormal distribution. Since in our examinations this condition isfulfilled, for all samples we have calculated the means and thestandarderrorsforneuronsofallthetestedlaminae.Anexampleof such mean values is presented in Fig. 1 by open circles alongwith the corresponding bars of standard errors.Consider five series of data as samples taken from definedareas. These samples consist of unequal numbers of variates.We can make the statistical hypothesis (H0) that all these sam-ples are taken from the same population. Since in this caselittle or no knowledge is available about the distribution ofthese five populations, it is advisable to employ methods thatdonotrequireassumptionsregardingthedatadistribution.Non-parametric statistics offers such methods. It is well known thatthe most popular nonparametric method taking into considera-tion all the samples together is the Kruskal–Wallis test [6]. Wehave applied this test to our five samples and found H=5.62.Since the calculated H-value did not exceed the tabular valueof H0.05=9.49 for degree of freedom (d.f.)=4, we have con-cluded that the result is not significant, and that the hypothesisH0is justified to the 5% level of significance. Thus, there are nosample pairs showing significantly different members.Testing a statistical hypothesis does not constitute a mathe-matical proof of the truth or falsity of the hypothesis. It seemsthat our statistical conclusion is incorrect and that in this casea type 1 error is committed [1]. The reason for this impressionseems to be a pronounced biological dispersion of data points(Fig. 1). To solve the problem, we have applied a technique ofpolynomial regression to the data obtained, in order to find the“best fit” function. To that purpose, we have solved the sys-tem of linear Eq. (5) and the values of constants ajreplaced inEq. (4). In Fig. 1 such a function is a polynomial of degree 4and the corresponding fitted curve is graphed as a dashed line.

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Author's personal copyD. Ristanovi´ c, N.T. Miloˇ sevi´ c / Neuroscience Letters 414 (2007) 286–290289Table 1Application of the Kolmogorov–Smirnov test to assess the significance of pairs of samples whose one member differs from the otherLaminae L1–L23L1–L4 L1–L5L1–L6 L23–L4L23–L5L23–L6 L4–L5L4–L6 L5–L6p>0.050.025 0.0090.0050.0310.019 0.009 >0.05>0.05 >0.05The difference between an observed value of N (given by theordinate of a point in Fig. 1) and the corresponding predictedN-value for the same r calculated from such a polynomial ofdegree 4 (not shown) represents the error of estimation. Theseerrors should be kept as small as possible. The requirement canbe satisfied if the degree of the polynomial is so high that thecoefficient of correlation R is almost equal to unity. The estima-tion errors become smaller but the curve becomes irregular. Wehave found the Lagrange’s interpolation polynomial of degree9, and the corresponding graph is presented in Fig. 1 as a fullline. This formula appears to be useless here since the graph ofthis function passes the experimental points (the coefficient ofcorrelation is equal 1) so that the sample of predicted N-values,used from this curve for a given series of consecutive values ofr (10, 25, 40, ..., ?m) is the same as that shown by the actual(or observed) N-values. It should be noted that there is no ruleon how to choose the degree of a polynomial of “best fit” thatwould allow for the best result: if the degree is small, the fittedcurve is rather regular, but it passes far from the experimen-tal points (the R is small). On the other hand, if the degree istoo high, the curve is irregular approaching the correspondinginterpolation polynomial. Using the curves of degrees between4 and 10 fitting the mean data for each lamina, we concludedthat it is best to use those for which the degree was 6. In thatcase the coefficient of correlation is 0.98 or more, except forthe lamina I where this value is 0.97. From the polynomialsgiven by Eq. (4)—adopting the corresponding constants ai, sep-arate plots have been constructed for each lamina and shownin Fig. 2.Fig.2. Thegraphsofpolynomialsfittedtodatapoints(notshown)correspondingto neurons from lamina I (L1), substantia gelatinosa (L2–3), lamina IV (L4),lamina V (L5) and lamina VI (L6). The graphs show the relationships betweenthe number of intersections N of dendrites with concentric circles against theradius r.After that we have calculated the values of N from thesepolynomial functions (4) using the series of data for r asbefore and applied again the Kruskal–Wallis test [6] tothese five samples. In this case, the value for H=10.08 andp=0.0391—showing a significant result. The extended mediantest showed χ2=15.04 and p=0.0046. As a post hoc test wehave used the Kolmogorov–Smirnov test for the unpaired sam-ple sizes [6]. Duan et al. [2] have used the data from the Shollanalysis as cumulative frequencies and statistical differencesbetween the two groups of their data assessed using also theKolmogorov–Smirnov test. The results of our testing are col-lected in Table 1 showing that there are 6 out of 10 possiblepairs of samples where members differs significantly, i.e. onelamina is significantly distinguishable from the other.Rexed’s laminar organization of the spinal grey matter hasbeen rapidly adopted by most neuroanatomists. But, it could bequestioned whether data obtained in the cat can be generalizedto other animals, since it has been noticed that Rexed’s laminarscheme can be found in all mammals, but not in other verte-brates [18]. Our quantitative morphologic analysis of dendriticarborization patterns of the cat spinal cord favors Rexed’s con-ceptofthelaminarscheme,sinceeachlaminaI–VIofthecatandother species testifies to a quite specific dendritic organization[9,14,15]. Gobel [4,5] has demonstrated that each of laminaeI–IV of the dorsal horn in the cat is dendroarchitectonicallydistinguishable.It should be noticed that the values taken over a heteroge-neous set of cells may be misleading and difficult to interpret.In such case the histograms or plots of the intersections withthe radii of concentric circles have similar features characteris-tic to the type of cell. Sholl has shown that the distributions ofthe N for stellate and pyramidal cells in the visual and motorcortices of the cat are generally similar; however, the distribu-tions show differences between the same areas [19]. This seemsto explain why has this analysis been used for many years andappliedtotheneuronsofthecerebralcortex[2].Indeed,itiswellknown that nearly all cortical neurons can be placed into one ofonlythreecategories:pyramidal,stellate(granular)andfusiform(bipolar) cells, the last ones being relatively rare. Besides, theyare arranged into six laminae that lie parallel to the pial surface.For instance, lamina II comprises mainly stellate cells, laminaIIIisanexternalpyramidallayermostlyconstitutedbythesmallsized pyramidal neurons, and lamina V is an internal pyramidallayer dominated by large pyramidal cells. It means that laminaIIcanbecompletelydescribedusingthemainpropertiesofstel-late cells while laminae III and V, using those of a pyramidalcell. On the other hand, all laminae of a dorsal horn consist ofmanydifferenttypesofneurons,suchasislet,filamentous,curly,stellate, antenna-like, stalked, arboreal, and fusiform (bipolar),limitrophe, radial etc. For instance, Schoenen [18] has reportedthat lamina II of the human dorsal horn contains: stellate cells